Superfast Coloring in CONGEST via Efficient Color Sampling

Summary

We present a procedure for efficiently sampling colors in the model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to semi-random colors unused by their neighbors in O(1) rounds, even in the distance-2 setting. This yields algorithms with complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. In particular, we obtain an -round algorithm for -edge coloring when , and a poly( )-round algorithm for -edge coloring in general. The sampling procedure is inspired by a seminal result of Newman in communication complexity.